Linear Algebra Section 4.4 Flashcards

Questions and Answers

What is a linear combination of vectors?

A set of vectors cannot be combined.

A vector can be expressed as a sum of scaled vectors. (correct)

A vector is always independent from others.

A vector can only be created from itself.

What is a spanning set of a vector space?

A set where every vector in the space can be represented as a linear combination of its vectors.

Define the span of a set.

The set of all linear combinations of the vectors in the set.

Span(S) is a subspace of V.

True Signup and view all the answers

When is a set of vectors called linearly independent?

When the only solution to the vector equation is the trivial solution. Signup and view all the answers

A set S is called linearly dependent if it has _____ solutions.

nontrivial Signup and view all the answers

What are the steps to test for linear independence?

Create a system of equations from the vector equation and check if it has a unique solution. Signup and view all the answers

A set is linearly independent if one of the vectors is a linear combination of others.

False Signup and view all the answers

Two vectors in a vector space are linearly independent if one is a scalar multiple of the other.

False Signup and view all the answers

Study Notes

Linear Combinations

A vector v in a vector space V is a linear combination of vectors u1, u2, ..., uk if it can be expressed as c1u1 + c2u2 + ... + ckuk, where c1, c2, ..., ck are scalars.

Spanning Sets

A subset S = {v1, v2, ..., vk} of a vector space V is a spanning set if every vector in V can be expressed as a linear combination of vectors in S.
If S spans V, it means that the entirety of V can be reached by taking linear combinations of vectors in S.

Span of a Set

The span of a set of vectors S = {v1, v2, ..., vk} is defined as the collection of all linear combinations of the vectors in S.
Denoted as span(S) or span{v1, v2, ..., vk}, it represents all possible linear combinations using real numbers as coefficients.
If span(S) = V, then it is stated that V is spanned by S.

Subspace Properties

Span(S) is a subspace of the vector space V that includes all linear combinations of vectors in S.
It is the smallest subspace containing S, meaning any other subspace that contains S must also include span(S).

Linear Dependence and Independence

A set of vectors S = {v1, v2, ..., vk} is linearly independent if the equation c1v1 + c2v2 + ... + ckvk = 0 only holds for the trivial solution c1 = 0, c2 = 0, ..., ck = 0.
If there are nontrivial solutions (other than the trivial one), then the set S is linearly dependent.

Testing Dependence/Independence

To test whether a set S is linearly independent or dependent:
- Set up the equation c1v1 + c2v2 + ... + ckvk = 0 and transform it into a system of linear equations.
- Apply Gaussian elimination to assess the uniqueness of the solution.
- If the solution is unique (only trivial), S is independent; if there are nontrivial solutions, it is dependent.

Theorem on Linear Independence

A set of vectors S = {v1, v2, ..., vk}, with k ≥ 2, is linearly independent only if at least one vector can be expressed as a linear combination of the others.

Corollary on Two Vectors

Two vectors u and v in a vector space V are linearly independent if and only if neither is a scalar multiple of the other.

Description

Test your knowledge on spanning sets and linear independence with these flashcards. Understand key concepts like linear combinations and spanning sets in vector spaces. Perfect for students studying linear algebra.